Do you have Windows on the machine? Try running some OpenGL in Windows (if you can) and see if your system is stable.
I had -exactly- the same problem with a Radion 9800 on another system, but only in Windows. It would run fine on the desktop, but run for random amounts of time before crashing so solidly that you had to use the reset button or poweroff when asked to do OpenGL. I finally convinced the vendor I bought the PC from that it was the Radeon, which they replaced under warranty and now everything is fine and dandy.
So I'd start by seeing if you have the same instability problems in Windows (if possible) while running at the same refresh rate and resolution in a Windows OpenGL game / application. If it -does- work here, reliably, Linux might be at fault. If it does fall over too, it must be the card - I have a strong suspicion you might have a bad card from your description of the fact that not even CTRL-ALT-DEL can elicit a response.
One more reason I won't ever buy ATI again...